Grothendieck-Verdier module categories, Frobenius algebras and relative Serre functors

We develop the theory of module categories over a Grothendieck-Verdier category $C$, i.e. a monoidal category with a dualizing object and hence a duality structure more general than rigidity. Such a category comes with two monoidal structures ⊗ and

which are related by non-invertible morphisms and which we treat on an equal footing. Quite generally, non-invertible structure morphisms play a dominant role in this theory. In any Grothendieck-Verdier module category $M$ we find two distinguished subcategories

and ${\hat{M}}^{\otimes }$, which can be characterized by certain structure morphisms being actually invertible. The internal Hom $A_m:=\underset{\_}{\mathrm{Hom}}(m,m)$ of an object m in ${\hat{M}}^{\otimes }$ that is a $C$-generator is an algebra such that $\mathrm{mod}\text{-}{A}_m$ is equivalent to $M$ as a module category. Crucially, the subcategories

and ${\hat{M}}^{\otimes }$ are precisely those on which a relative Serre functor can be defined. This relative Serre functor furnishes an equivalence

, and any isomorphism $m\stackrel{\cong }{\to }S\left(m\right)$ endows the algebra $A_m$ with the structure of a Grothendieck-Verdier Frobenius algebra.